Properties

Degree $2$
Conductor $7056$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82·5-s + 6·11-s + 5.65·13-s + 1.41·17-s − 4.24·19-s + 4·23-s + 3.00·25-s + 6·29-s + 2.82·31-s + 2·37-s − 1.41·41-s − 10·43-s + 2.82·47-s + 2·53-s + 16.9·55-s − 1.41·59-s + 8.48·61-s + 16.0·65-s − 4·67-s − 12·71-s + 9.89·73-s + 4·79-s − 1.41·83-s + 4.00·85-s − 4.24·89-s − 12·95-s − 12.7·97-s + ⋯
L(s)  = 1  + 1.26·5-s + 1.80·11-s + 1.56·13-s + 0.342·17-s − 0.973·19-s + 0.834·23-s + 0.600·25-s + 1.11·29-s + 0.508·31-s + 0.328·37-s − 0.220·41-s − 1.52·43-s + 0.412·47-s + 0.274·53-s + 2.28·55-s − 0.184·59-s + 1.08·61-s + 1.98·65-s − 0.488·67-s − 1.42·71-s + 1.15·73-s + 0.450·79-s − 0.155·83-s + 0.433·85-s − 0.449·89-s − 1.23·95-s − 1.29·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{7056} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.550358959\)
\(L(\frac12)\) \(\approx\) \(3.550358959\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 2.82T + 5T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 - 1.41T + 17T^{2} \)
19 \( 1 + 4.24T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 2.82T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 1.41T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 1.41T + 59T^{2} \)
61 \( 1 - 8.48T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 9.89T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 1.41T + 83T^{2} \)
89 \( 1 + 4.24T + 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.186606903107144253527389982554, −6.86449818490525449453570564035, −6.48807244167066257063252337390, −6.04388864948741487810013134854, −5.22714853778186210804765791226, −4.27212020799128708050598395436, −3.62953030193807424433779539844, −2.66961043669717636699923991565, −1.57661125791959474201251567119, −1.11981705315670534212033177055, 1.11981705315670534212033177055, 1.57661125791959474201251567119, 2.66961043669717636699923991565, 3.62953030193807424433779539844, 4.27212020799128708050598395436, 5.22714853778186210804765791226, 6.04388864948741487810013134854, 6.48807244167066257063252337390, 6.86449818490525449453570564035, 8.186606903107144253527389982554

Graph of the $Z$-function along the critical line